Deep learning and the Schrödinger equation

@article{Mills2017DeepLA,
  title={Deep learning and the Schr{\"o}dinger equation},
  author={Kyle Mills and Michael A. Spanner and Isaac Tamblyn},
  journal={ArXiv},
  year={2017},
  volume={abs/1702.01361}
}
A deep (convolutional) neural network is trained to predict the ground-state energy of an electron in two-dimensional potentials. The machinery of deep learning is developed to learn the mapping between potential and energy, which bypasses the need to numerically solve the Schr\"odinger equation and the need for computing wave functions. 

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References

SHOWING 1-10 OF 153 REFERENCES
Modeling electronic quantum transport with machine learning
TLDR
The remarkable performance of the machine learning approach to solve electronic quantum transport equations of one-dimensional nanostructures to capture the complexity of interference phenomena lends further support to its viability in dealing with transport problems of undulatory nature.
Machine learning for many-body physics: The case of the Anderson impurity model
Machine learning methods are applied to finding the Green's function of the Anderson impurity model, a basic model system of quantum many-body condensed-matter physics. Different methods of
Kinetic Energy of Hydrocarbons as a Function of Electron Density and Convolutional Neural Networks.
TLDR
A convolutional neural network trained to reproduce the Kohn-Sham kinetic energy of hydrocarbons from an input electron density is demonstrated and it is shown that this approximation qualitatively reproduces KOHN-Sham potential energy surfaces when used with conventional exchange correlation functionals.
Finding Density Functionals with Machine Learning
TLDR
For the model problem of the kinetic energy of noninteracting fermions in 1D, mean absolute errors below 1 kcal/mol on test densities similar to the training set are reached with fewer than 100 training densities.
Solving the quantum many-body problem with artificial neural networks
TLDR
A variational representation of quantum states based on artificial neural networks with a variable number of hidden neurons and a reinforcement-learning scheme that is capable of both finding the ground state and describing the unitary time evolution of complex interacting quantum systems.
Fast and accurate modeling of molecular atomization energies with machine learning.
TLDR
A machine learning model is introduced to predict atomization energies of a diverse set of organic molecules, based on nuclear charges and atomic positions only, and applicability is demonstrated for the prediction of molecular atomization potential energy curves.
Quantum-chemical insights from deep tensor neural networks
TLDR
An efficient deep learning approach is developed that enables spatially and chemically resolved insights into quantum-mechanical observables of molecular systems, and unifies concepts from many-body Hamiltonians with purpose-designed deep tensor neural networks, which leads to size-extensive and uniformly accurate chemical space predictions.
Using neural networks to solve nonlinear differential equations in atomic and molecular physics
To represent the solution of a differential equation by an artificial neural network (ANN) was an idea introduced by Lagaris. Sugawara applied this concept to solve Schrodinger's equation for select
...
...